IB Maths AI HL · Unit 4: Statistics and Probability
IB Maths AI HL Random Variables and Poisson Questions
Exam-style IB Maths AI HL random variables and poisson questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 17 questions
- Paper 1: 17
- 5 easy
- 5 medium
- 5 hard
- 2 starter
- 3 worked examples
Practise Random Variables and Poisson questions →
AI HL formula booklet
What's examined in AI HL random variables and poisson
The question bank covers these random variables and poisson question types (number of questions in brackets):
- Poisson Distribution Modelling (9)
- Expected Value and Variance Rules (7)
- Probability and Distribution Properties (1)
Key formulas
- Mean & variance of Poisson
- \(E(X) = \lambda,\ \mathrm{Var}(X) = \lambda\)
- Poisson distribution
- \(P(X = r) = \dfrac{e^{-\lambda}\lambda^{\,r}}{r!}\)
In the same notation as the IB formula booklet. All AI HL formulas →
Random Variables and Poisson worked examples
Worked example 1: Calculating Basic Poisson Probabilities · easy
The number of emails received by a small business support desk follows a Poisson distribution with an average rate of $4.5$ emails per hour. Find the probability that the desk receives exactly $6$ emails in a randomly chosen hour.
1. Define the random variable $X$ as the number of emails per hour. $X \sim Po(4.5)$.
2. Set up the probability statement: $P(X = 6)$.
3. Use the Poisson Probability Density (Ppd) function on your GDC with $\lambda = 4.5$ and $x = 6$.
4. Alternatively, substitute into the formula: $P(X = 6) = \frac{e^{-4.5} \times 4.5^6}{6!}$.
5. Evaluate the probability: $0.12812\dots$
6. The probability is $0.128$ (to 3 s.f.).
Examiner tip: Use `Poisson Ppd` (Point) for exact values like $P(X=6)$, and use `Poisson Pcd` (Cumulative) for range values like $P(X \le 6)$.
Worked example 2: Scaling the Poisson Mean · medium
A factory machine breaks down at an average rate of $0.8$ times per week. Assuming the breakdowns follow a Poisson distribution, calculate the probability that the machine breaks down exactly $4$ times in a $4$-week period.
1. Identify the original rate: $\lambda = 0.8$ breakdowns per $1$ week.
2. Scale the mean proportionally for the new time period ($4$ weeks): New mean $m = 0.8 \times 4 = 3.2$ breakdowns.
3. Define the new random variable $Y$ for the $4$-week period: $Y \sim Po(3.2)$.
4. Set up the required probability: $P(Y = 4)$.
5. Evaluate using the GDC Poisson Ppd tool with $\lambda = 3.2$ and $x = 4$.
6. The probability is $0.17809\dots$, giving $0.178$.
Examiner tip: The defining characteristic of the Poisson distribution is that the mean ($\lambda$) is strictly proportional to the size of the interval (time, area, or volume) being observed.
Worked example 3: Summing Independent Poisson Distributions · hard
On a quiet rural road, the number of cars passing per hour follows a Poisson distribution with mean $2.5$. The number of trucks passing per hour follows an independent Poisson distribution with mean $1.2$. Find the probability that a total of exactly $5$ vehicles (cars and trucks combined) pass by in a randomly chosen one-hour period.
1. Recall the reproductive property of the Poisson distribution: the sum of independent Poisson variables is also a Poisson variable.
2. Define the combined random variable $V = C + T$.
3. Calculate the new combined mean: $\lambda_V = 2.5 + 1.2 = 3.7$.
4. Establish the new distribution: $V \sim Po(3.7)$.
5. Set up the probability statement for the total vehicles: $P(V = 5)$.
6. Evaluate using the GDC Poisson Ppd tool or the formula $\frac{e^{-3.7} \times 3.7^5}{5!}$.
7. The exact probability is $0.14287\dots$, giving $0.143$.
Examiner tip: You can only add the means ($\lambda$) to form a new Poisson distribution if the individual events are completely independent of each other.
Try these IB Maths AI HL random variables and poisson questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Question 1 · easy · 4 marks · Paper 1
A random variable \(X\) has an expected value \(E(X) = 8\) and a variance \(Var(X) = 5\).
A new variable is defined by the linear transformation \(Y = 3X - 4\).
Calculate \(E(Y)\) and \(Var(Y)\).
Question 2 · medium · 5 marks · Paper 1
The weights of apples are normally distributed with a mean of \(150\)g and a standard deviation of \(10\)g.
If two apples are chosen at random, find the probability that their combined weight is greater than \(315\)g.
Question 3 · hard · 7 marks · Paper 1
Let \(X\) be the volume of juice in a bottle, where \(E(X) = 500\)ml and \(Var(X) = 15\).
A student claims that taking one bottle and multiplying its volume by two, \(Y = 2X\), has the exact same variance as taking two independent bottles and adding their volumes together, \(W = X_1 + X_2\).
Determine algebraically whether the student is correct, and explain the mathematical reason for the difference in variance.
All 17 random variables and poisson questions with mark schemes →
FAQ
How many IB Maths AI HL random variables and poisson questions are there?
There are 17 exam-style random variables and poisson questions in the AI HL question bank (Paper 1: 17), graded 5 easy, 5 medium, 5 hard, 2 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is random variables and poisson on Paper 1 or Paper 2?
In the question bank these questions are set as Paper 1 questions.
Where can I get the mark schemes?
Open the AI HL Unit 4 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.
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